Projective Ring Line of a Specific Qudit

TL;DR

This paper explores the relationship between the commutation relations of generalized Pauli operators for a specific class of qudits and the structure of the projective line over modular rings, revealing new geometric insights.

Contribution

It establishes a novel connection between qudit operator commutation and projective geometry over rings, including the role of non-admissible pairs in this framework.

Findings

Operators correspond to points on a projective line over rac{rac{d}{d}

Perp-sets form set-theoretic unions of projective line points

Non-admissible pairs expand the geometric representation of operators

Abstract

A very particular connection between the commutation relations of the elements of the generalized Pauli group of a $d$-dimensional qudit, $d$ being a product of distinct primes, and the structure of the projective line over the (modular) ring $\bZ_{d}$ is established, where the integer exponents of the generating shift ($X$) and clock ($Z$) operators are associated with submodules of $\bZ^{2}_{d}$. Under this correspondence, the set of operators commuting with a given one -- a perp-set -- represents a $\bZ_{d}$-submodule of $\bZ^{2}_{d}$. A crucial novel feature here is that the operators are also represented by {\it non}-admissible pairs of $\bZ^{2}_{d}$. This additional degree of freedom makes it possible to view any perp-set as a {\it set-theoretic} union of the corresponding points of the associated projective line.